A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra B in a braided monoidal category \({{\mathcal C}}\), and under certain assumptions on the braiding (fulfilled if \({{\mathcal C}}\) is symmetric), we construct a sequence for the Brauer group \({{\rm{BM}}}({{\mathcal C}};B)\) of B-module algebras, generalizing Beattie’s one. It allows one to prove that \({{\rm{BM}}}({{\mathcal C}};B) \cong {{\rm{Br}}}({{\mathcal C}}) \times {\operatorname{Gal}}({{\mathcal C}};B)\), where \({{\rm{Br}}}({{\mathcal C}})\) is the Brauer group of \({{\mathcal C}}\) and \({\operatorname{Gal}}({{\mathcal C}};B)\) the group of B-Galois objects. We also show that \({{\rm{BM}}}({{\mathcal C}};B)\) contains a subgroup isomorphic to \({{\rm{Br}}}({{\mathcal C}}) \times {\operatorname{H^2}}({{\mathcal C}};B,I),\) where \({\operatorname{H^2}}({{\mathcal C}};B,I)\) is the second Sweedler cohomology group of B with values in the unit object I of \({{\mathcal C}}\). These results are applied to the Brauer group \({{\rm{BM}}}(K,B \times H,{{\mathcal R}})\) of a quasi-triangular Hopf algebra that is a Radford biproduct B × H, where H is a usual Hopf algebra over a field K, the Hopf subalgebra generated by the quasi-triangular structure \({{\mathcal R}}\) is contained in H and B is a Hopf algebra in the category \({}_H{{\mathcal M}}\) of left H-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that \({{\rm{BM}}}(K,H,{{\mathcal R}}) \times {\operatorname{H^2}}({}_H{{\mathcal M}};B,K)\) is a subgroup of \({{\rm{BM}}}(K,B \times H,{{\mathcal R}})\), confirming the suspicion that a certain cohomology group of B × H (second lazy cohomology group was conjectured) embeds into it. New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.